The Selberg Zeta Function for Convex CoCompact Schottky Groups
Abstract
We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $ \exp (C s^\delta) $ where $ \delta $ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound $ \exp (C s^{n+1}) $, and it gives new bounds on the number of resonances (scattering poles) of $ \Gamma \backslash {\mathbb H}^{n+1} $. The proof of this result is based on the application of holomorphic $ L^2$techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider $ \Gamma \backslash {\mathbb H}^{n+1} $ as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic $L^2$techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2004
 DOI:
 10.1007/s0022000310071
 arXiv:
 arXiv:math/0211041
 Bibcode:
 2004CMaPh.245..149G
 Keywords:

 Differential Geometry;
 Mathematical Physics;
 Spectral Theory;
 37C30;
 11M36;
 37F30;
 30F40;
 37M25
 EPrint:
 Communications in Mathematical Physics 245 (2004) 149176